Contributions to Discrete Mathematics (E-Journal, University of Calgary)
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Moments of -Jacobi polynomials and -zeta values
Full text linkWe explore some connections between moments of rescaled little -Jacobi polynomials, -analogues of values at negative integers for some Dirichlet series, and the -Eulerian polynomials of wreath products of symmetric groups
On emulational equivalence of impartial games and the game Hackenforb
Full text linkWe introduce a variant of the game Hackenbush, called Hackenforb. It is a class of games, each of which is determined by two parameters: a given graph, and a given set of connected graphs (called forbidden graphs). The significance of this game within the realm of impartial combinatorial games is reflected in the fact that, as we show in this article, various known combinatorial games, such as Nim, Subtraction game, Notakto, Treblecross, Chomp, are emulationally equivalent to an instance of Hackenforb (an emulational equivalence of two games is a concept stronger than Grundy-equivalence, but weaker than the isomorphism between games\u27 structures; our belief is that this version of equivalence is what really captures the core of the intuitive perception of what it means for two games to be ``basically the same game"). At the end of our article, we show that Hackenforb is, unfortunately, not ``almighty," that is, we describe a game that is not emulationally equivalent to an instance of Hackenforb
A bijection between the sets of -generalized Motzkin paths avoiding -patterns and -patterns
No full textA generalized Motzkin path, called G-Motzkin path for short, of length is a lattice path from to in the first quadrant of the XY-plane that consists of up steps , horizontal steps , vertical steps and down steps . An -G-Motzkin path is a weighted G-Motzkin path such that the -steps, -steps, -steps and -steps are weighted respectively by and .Let be a word on , denote by the set of -avoiding -G-Motzkin paths of length for a pattern . In this paper, we consider the -avoiding -G-Motzkin paths and provide a direct bijection between and . Finally, the set of fixed points of is also described and counted
Enumeration of parallelogram polycubes
Full text linkIn this paper, we give the Dirichlet generating function of parallelogram polycubes according to the volume and the width in terms of multivariate zeta functions. We also enumerate them according to the width, the length and the depth. All these results are generalized to polyhypercubes
Fair partitions of the plane into incongruent pentagons
Full text linkMotivated by a question of R. Nandakumar, we show that the Euclidean plane can be dissected into mutually incongruent convex pentagons of the same area and the same perimeter
The flag - and - vectors of generalized Square posets
Full text linkTiling a quadrant of the plane with squares gives rise to the Square poset. Adjustments in the tiling tactic generate a series of posets that we refer to as the generalized Square posets. We study the flag - and -vectors of this class of generalized Square posets
Saved by the rook: a case of matchings and Hamiltonian cycles
Full text linkThe rook graph is a graph whose edges represent all the possible legal moves of the rook chess piece on a chessboard. The problem we consider is the following. Given any set containing pairs of cells such that each cell of the chessboard is in exactly one pair, we determine the values of the positive integers and for which it is possible to construct a closed tour of all the cells of the chessboard which uses all the pairs of cells in and some edges of the rook graph. This is an alternative formulation of a graph-theoretical problem presented in Marién Abreu, John Baptist Gauci, Domenico Labbate, Giuseppe Mazzuoccolo, and Jean Paul Zerafa, Extending perfect matchings to Hamiltonian cycles in line graphs, Electron. J. Comb. 28 (2021), no.~1, research paper p1.7, 13 (English) involving the Cartesian product of two complete graphs and , which is, in fact, isomorphic to the rook graph. The problem revolves around determining the values of the parameters and that would allow any perfect matching of the complete graph on the same vertex set of to be extended to a Hamiltonian cycle by using only edges in
Reversed Dickson polynomials of the (k+1)-th kind over finite fields, II
Full text linkLet be an odd prime. In this paper, we study the permutation behaviour of the reversed Dickson polynomials of the -th kind when , , and , where , , and are nonnegative integers. A generalization to is also shown. We find some conditions under which is not a permutation polynomial over finite fields for certain values of and . We also present a generalization of a recent result regarding and present some algebraic and arithmetic properties of
Colourings of aperiodic tilings
Full text linkWe find explicit optimal vertex, edge and face coulourings for the chair tiling, the Ammann--Beenker tiling, the rational pinwheel tiling and the pinwheel tiling
Covering the crosspolytope with crosspolytopes
Full text linkLet be the smallest positive number such that the convex body can be covered by translates of . Let be the -dimensional crosspolytope. It will be proved that for 1\le m< 2d, ; for , ; for , ; for , and for , . Moreover the Hadwiger’s covering conjecture is verified for the -dimensional crosspolytope